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4 Answers
4

For counting a number of occurrences 回 and 度 are interchangeable with small numbers. Somewhere around 4 (the line is quite vague), 度 becomes uncommon, and by the time you get to 6, 回 is pretty much the only one used. (Naturally, 度 can be used with any number for counting degrees, as noted in Azeworai's informative answer.)

Note: I'm posting this as answer rather than adding it to the question for two reasons: (1) I came up with this theory after I posted the question and after I read the other answers (2) adding this to the question would significantly change the nature of the question and it'd be unfair to those who already answered

After reading answers from Derek and Ross, I came up with a theory on how 度 and 回 differ from each other. Even though this theory came from my observation and lacked reference, the more I think about it the more probable it becomes so it might be the case that it is actually true.

Looking at the kanji for 回, it's the same kanji that is used for 回る, 回数, 回転 etc, which all imply a circular flow, a rotation, a repetition. This nuance of repetition is what 回 has and 度 lacks. The notion of repeating event brings about another nuance: intention (if you are causing the event to happen) and expectation (if you are just observing it). When an event is known to repeat, there is an expectation that it will happen again, while an event that simply occurs may not be expected to repeat.

If this is actually true, then it would explain Derek's answer why 度 becomes uncommon after the 4th time (or whatever the vague limit is). Events may occur the first time, second time and third time, but after the fourth time it is hard not to expect it will happen the fifth time. The fifth time is no longer "yet another occurrence"; there is an expectation for it to happen; the fifth time and so on are now repetitions.

On the other hand, 度 which is also used as a counter for degrees, while may not signify the notion of repetition, it has the notion of progression, i.e. that we do not only look at the occurrences as events that happen again and again but also compare the quality/quantity of the occurrences among each other. While 回 focuses on repetitions, 度 focuses on progressions. This can be seen from the subtle difference between 「今回」 and 「今度」.

This difference between 度 and 回 also extends to the choice between using 「もう一回」 and 「もう一度」 in a request. Based on my observation, 「もう一回」 in a request is loaded with intention and expectation that the requested action will happen, while 「もう一度」 lacks that expectation so it makes the request rather neutral and reserved. This probably explain why 「もう一回」 sounds a bit childish and is commonly used by children due to its bluntness in showing how they expect and insist the request to be fulfilled. On the other hand, the notion of progression in 度 loads 「もう一度」 with a request to improve the action. That's probably why when we want to ask somebody to repeat what he says because we couldn't hear the first time, it's more natural to use 「もう一度言ってください」 because we want the speaker to say it better/louder. 「もう一回言ってください」 may sound like we want the speaker to simply repeat exactly what he said the first time.

At this point, let's further extend this theory to distinguish the example sentences in my question. I think it's no farfetched to say that the difference between using 度 and 回 in saying "I haven't done it even once" lies in your expectation and intention; whether you want to do it or not):

そんなことは一回もしたことがない
I haven't done such thing even once (and I want to do it).

そんなことは一度もしたことがない
I haven't done such thing even once (and I do not even want to do it).

Your theory is interesting, but I am not sure if it matches the reality. I do not recognize the difference in nuance in your last two examples (but I am not always aware of the nuance of expressions). Moreover, a proverb 二度あることは三度ある (if something happens twice, it happens three times; things often repeat) is literally about repetition of an event, but it uses 度. But right now I do not see a strong evidence for or against your theory.
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Tsuyoshi ItoSep 8 '11 at 23:11

I also don't consider there to be any difference in nuance. Your point about cyclic motion with 回 seems valid enough, but 大辞泉 defines 度数 as nothing more than 回数. It's an interesting idea, but I don't think you can necessarily jump from a claim about repetition to a claim about volition. If we're talking characters, the 'every time X' construction ～度(たび) certainly doesn't hinge on intention/expectation. (And while we're counting occurrences, there's technically also 次, which has limited use in Japanese (第二次世界大戦, 二次会, etc.), but is the basic counter for occurrences in Chinese.)
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Ross KirslingSep 8 '11 at 23:30

@Lukman - I have to admit that I'm not too keen on grand unifying theories of this kind, but you might consider this useful to consider. 回, meaning revolution, implies periodicity, whereas 度, meaning measurement, implies gradation. Whether that's significant or not, I haven't a clue.
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rdbSep 9 '11 at 4:03

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@Lukman: (1) You may be right about 二度あることは三度ある. If we do not count it as an (admittedly already weak) evidence against your theory, I do not have anything objective. (2) I still just do not feel that the theory matches the reality. I do not think that もう一度 and もう一回 have any difference in whether the second time is completely the same as the first time or not. And, as I said, I do not feel the difference which you described between 一回もない and 一度もない. 今度 and 今回 can be used very differently, but I am not sure if the difference is well explained by your theory.
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Tsuyoshi ItoSep 9 '11 at 12:22